Formalising foundations of mathematics

1992

Abstract Zermelo-Fraenkel (ZF) set theory is widely regarded as unsuitable for automated reasoning. But a computational logic has been formally derived from the ZF axioms using Isabelle. The library of theorems and derived rules, with Isabelle's proof tools, support a natural style of proof. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge [3], and Ramsey's Theorem [2].

We present a programme of research for pluralist formalisations — formalisations that involve proving results in more than one foundation. A foundation consists of two parts: a logical part that provides a notion of inference, and a non-logical part that provides the entities to be reasoned about. A logic-enriched type theory (LTT) is a formal system composed of such two separate parts. We show how LTTs may be used as the basis for a pluralist formalisation. We show how different foundations may be formalised as LTTs, and also describe a new method for proof reuse. If we know that a translation Φ exists between logic-enriched type theories (LTTs) S and T , and we have formalised a proof of a theorem α in S, we may wish to make use of the fact that Φ(α) is a theorem of T . We show how this is sometimes possible by writing a proof script MΦ . For any proof script Mα that proves a theorem α in S, if we change Mα so it first imports MΦ , then the resulting proof script will still parse, and will be a proof of Φ(α) in T . In this paper, we focus on the logical part of an LTT-framework and show how the above method of proof reuse is done for four cases of Φ: inclusion, the double negation translation, the A-translation, and the Russell-Prawitz modality. This work has been carried out using the proof assistant Plastic.

2004

Mathematical Structures in Computer Science

This special issue of Mathematical Structures in Computer Science is devoted to the theme of 'Interactive theorem proving and the formalisation of mathematics'.

2012

A strictly formal, set-theoretical treatment of classical first-order logic is given. Since this is done with the goal of a concrete Mizar formalization of basic results (Lindenbaum lemma; Henkin, satisfiability, completeness (Goedel) and Lowenheim-Skolem theorems) in mind, it turns into a systematic pursue of simplification: we give up the notions of free occurrence, of derivation tree, and study what inference rules are strictly needed to prove the mentioned results. Afterwards, we discuss details of the actual Mizar implementation, and give general techniques developed therein.

The V language is based on ZFC set theory and intended for use mostly in education. So every proof step is quite simple and can be checked by a human. The most important syntactic construct in V is the class term. Class terms are used for defining theories, the lambda, set builder, and restricted quantification notations. For testing the V language the following 3 modules (modules in V are similar to articles in Mizar) were written: ❼ root: elementary set theory: 5134 lines, 143 proofs, 47 equational proofs; some of the theorems were declared as axioms, they have to be proved later; ❼ group: the basics of group theory, including the isomorphism theorems: 5402 lines, 508 proofs, 326 equational proofs; ❼ altg: the basics of the permutation theory, necessary for defining the alternating group: about 1700 lines, 136 proofs, 60 equational proofs; Also, a computer program (a proof checker) has been written(26300 lines in C++), which now is working in an experimental mode. The 3 modules have been successfully checked by the proof checker. The processing times for the root, group, and altg modules are 35sec,53sec, and 14sec respectively on the laptop Lenovo ThinkPad Yoga 11e 5th Gen. The modules can be downloaded at [2].

Bulletin of Symbolic Logic, 2001

We discuss the differences between first-order set theory and secondorder logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former. * I am grateful to Juliette Kennedy for many helpful discussions while developing the ideas of this paper. † Research partially supported by grant 40734 of the Academy of Finland.

2003

In The Foundations of Mathematics in the Theory of Sets, John Mayberry attacks the 2000-year-old problem of accounting for the foundations of mathematics. His account comes in three interrelated parts: determining exactly what one should (and should not) expect from a foundation; arguing that set theory can in fact provide such a foundation, and presenting a novel version of set theory (or at least a novel exposition of traditional set theory) which can fulfil this foundational role.

2010

We propose a Relational Calculus based on the concept of unary relation. In this Relational Calculus different axiomatic systems converge to a model called Dynamic Generative System with Symmetry (DGSS). In DGSS we define the concepts of relational set and function and prove that extensionality and the substitution property of equality are theorems of DGSS. As a first exemplification of DGSS, we construct a model of natural numbers without relying on Peano's Axioms. Eventually, some new clarifications regarding the nature of the number zero are given.

The two main views in modern constructive mathematics, usually associated with constructive type theory and topos theory, are compatible with the classical view, but they are incompatible with each other, in a sense explained by some specific results which we briefly review. So it is desirable to design a common core which is compatible with all the theories in which mathematics has been developed, like Zermelo–Fraenkel set theory, topos theory, Martin-Löf's type theory, etc. and can be understood as it is by any mathematician, whatever foundation is adopted. A requirement with increasing importance is that of developing mathematics in such a way that it can be formalized on a computer. This theoretically means that the foundation should obey the proofs-as-programs paradigm. We claim that to satisfy both requirements it is necessary to use a minimal type theory mTT, which is obtained from Martin-Löf's type theory by relaxing the identification of propositions with sets. This ground type theory mTT is intensional and is needed for formalization. A 'toolbox' of extensional concepts, needed to do mathematics , is built on it. The common core is obtained at this level, by subtraction. The underlying conceptual novelty is that one should give up to the expectation of an all-embracing foundation, also in the concrete sense that one needs a formal system living at two different levels of abstraction. After abandoning the classical view and all its limits, a prospective constructivist is faced with the problem of choosing among a variety of views. They generally share the choice for intuitionistic logic, but they differ in mathematical principles. The principal distinction is between two views. One maintains that the meaning of mathematics lies in its computational content, and thus keeps its formalization in a computer language in mind. It is usually associated with type theory; the axiom of choice then turns out to be valid and (hence) the powerset axiom is not accepted [29]. The other favours the mathematical structure beyond its particular presentations. It is usually expressed through category theory and often identified with topos theory. Extensionality is an essential feature, the powerset axiom is mostly accepted and hence the axiom of choice is not accepted as valid [23]. Both views are reasonable, well motivated and apparently cannot be dispensed with. It is however a matter of fact that they are incompatible, in the sense that by accepting both one is forced back to the classical view. The same tension is revealed also as a technological challenge: while implementation of mathematics in a computer is intrinsically intensional (just think of the fact that